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Given the vector v has an initial point at (4,4) and a terminal point at (1,7), find the exact value of ||v||. Answer:

Given the vector v \mathbf{v} has an initial point at (4,4) (4,4) and a terminal point at (1,7) (1,7) , find the exact value of ∥v∥ \|\mathbf{v}\| .\newlineAnswer:

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Full solution

Q. Given the vector v \mathbf{v} has an initial point at (4,4) (4,4) and a terminal point at (1,7) (1,7) , find the exact value of ∥v∥ \|\mathbf{v}\| .\newlineAnswer:
  1. Use Distance Formula: To find the magnitude of the vector vv, we need to use the distance formula, which is derived from the Pythagorean theorem. The distance formula for a vector with initial point (x1,y1)(x_1, y_1) and terminal point (x2,y2)(x_2, y_2) is:\newline∣∣v∣∣=((x2−x1)2+(y2−y1)2)||v|| = \sqrt{((x_2 - x_1)^2 + (y_2 - y_1)^2)}
  2. Substitute Given Points: Substitute the given points into the distance formula:\newlineInitial point (x1,y1)=(4,4)(x_1, y_1) = (4, 4)\newlineTerminal point (x2,y2)=(1,7)(x_2, y_2) = (1, 7)\newline∣∣v∣∣=(1−4)2+(7−4)2||v|| = \sqrt{(1 - 4)^2 + (7 - 4)^2}
  3. Calculate Differences and Squares: Calculate the differences and square them:\newline∣∣v∣∣=(−3)2+(3)2||v|| = \sqrt{(-3)^2 + (3)^2}
  4. Simplify Squares: Simplify the squares:\newline∣∣v∣∣=9+9||v|| = \sqrt{9 + 9}
  5. Add Values Inside Square Root: Add the values inside the square root:\newline∣∣v∣∣=(18)||v|| = \sqrt{(18)}
  6. Simplify Square Root: Simplify the square root if possible. Since 1818 is not a perfect square, we leave it as 18\sqrt{18}, which is the exact value of the magnitude of vector vv.

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